Semi-empirical dissipation source functions for ocean
waves: Part I, definition, calibration and validation
Fabrice Ardhuin, Erick Rogers, Alexander Babanin, Jean-François Filipot,
Rudy Magne, Aron Roland, Andre Van Der Westhuysen, Pierre Queffeulou,
Jean-Michel Lefevre, Lotfi Aouf, et al.
To cite this version:
Fabrice Ardhuin, Erick Rogers, Alexander Babanin, Jean-François Filipot, Rudy Magne, et al..
Semi-empirical dissipation source functions for ocean waves: Part I, definition, calibration and
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Semi-empirical dissipation source functions for ocean waves:
Part I, definition, calibration and validation.
Fabrice Ardhuin ∗, Jean-François Filipot and Rudy Magne
Service Hydrographique et Océanographique de la Marine, Brest, France
Erick Rogers
Oceanography Division, Naval Research Laboratory, Stennis Space Center, MS, USA
Alexander Babanin
Swinburne University, Hawthorn, VA, Australia
Pierre Queffeulou
Ifremer, Laboratoire d’Océanographie Spatiale, Plouzané, France
Lotfi Aouf and Jean-Michel Lefevre
UMR GAME, Météo-France - CNRS, Toulouse, France
Aron Roland
Technological University of Darmstadt, Germany
Andre van der Westhuysen
Deltares, Delft, The Netherlands
Fabrice Collard
CLS, Division Radar, Plouzané, France
ABSTRACT
New parameterizations for the spectral dissipation of wind-generated waves are proposed. The rates
of dissipation have no predetermined spectral shapes and are functions of the wave spectrum, in a
way consistent with observation of wave breaking and swell dissipation properties. Namely, swell
dissipation is nonlinear and proportional to the swell steepness, and wave breaking only affects
spectral components such that the non-dimensional spectrum exceeds the threshold at which waves
are observed to start breaking. An additional source of short wave dissipation due to long wave
breaking is introduced, together with a reduction of wind-wave generation term for short waves,
otherwise taken from Janssen (J. Phys. Oceanogr. 1991). These parameterizations are combined
and calibrated with the Discrete Interaction Approximation of Hasselmann et al. (J. Phys. Oceangr.
1985) for the nonlinear interactions. Parameters are adjusted to reproduce observed shapes of
directional wave spectra, and the variability of spectral moments with wind speed and wave height.
The wave energy balance is verified in a wide range of conditions and scales, from the global ocean
to coastal settings. Wave height, peak and mean periods, and spectral data are validated using in
situ and remote sensing data. Some systematic defects are still present, but the parameterizations
probably yield the most accurate overall estimate of wave parameters to date. Perspectives for
further improvement are also given.
sities of the surface elevation variance F distributed over
frequencies f and directions θ can be put in the form
1. Introduction
a. On phase-averaged models
dF (f, θ)
= Satm (f, θ)+Snl (f, θ)+Soc (f, θ)+Sbt (f, θ), (1)
dt
Spectral wave modelling has been performed for the last
50 years, using the wave energy balance equation (Gelci
et al. 1957). This model for the evolution of spectral den-
1
where the Lagrangian derivative is the rate of change of
the spectral density when following a wave packet at its
group speed in physical and spectral space. This spectral
advection particularly includes changes in direction due to
the Earth sphericity and refraction over varying topography (e.g. Munk and Traylor 1947; Magne et al. 2007) and
currents, and changes in wavelength or period in similar
conditions (Barber 1949).
The source functions on the right hand side are separated into an atmospheric source function Satm , a nonlinear scattering term Snl , an ocean source Soc , and a bottom
source Sbt .
This separation, like any other, is largely arbitrary. For
example, waves that break are highly nonlinear and thus
the effect of breaking waves that is contained in Soc is intrisically related to the non-linear evolution term contained
in Snl . Yet, compared to the usual separation of deep-water
evolution into wind input, non-linear interactions, and dissipation, it has the benefit of identifying where the energy
and momentum is going to or coming from, which is a necessary feature when ocean waves are used to drive or are
coupled with atmospheric or ocean circulation models (e.g.
Janssen et al. 2004; Ardhuin et al. 2008b).
Satm , which gives the flux of energy from the atmospheric non-wave motion to the wave motion, is the sum
of a wave generation term Sin and a wind generation term
Sout (often referred to as “negative wind input”, i.e. a
wind output). The nonlinear scattering term Snl represents all processes that lead to an exchange of wave energy
and momentum between the different spectral components.
In deep and intermediate water depth, this is dominated
by cubic interactions between quadruplets of wave trains,
while quadratic nonlinearities play an important role in
shallow water (e.g. WISE Group 2007). The ocean source
Soc may accomodate wave-current interactions1 and interactions of surface and internal waves, but it will be here
restricted to wave breaking and wave-turbulence interactions.
The basic principle underlying equation (1) is that waves
essentially propagate as a superposition of almost linear
wave groups that evolve on longer time scales as a result of
weak-in-the-mean processes (e.g. Komen et al. 1994). Recent reviews have questioned the possibility of further improving numerical wave models without changing this basic
principle (Cavaleri 2006). Although this may be true in the
long term, we demonstrate here that it is possible to improve model results significantly by including more physical
features in the source term parameterizations. The main
advance proposed in the present paper is the adjustment
of a shape-free dissipation function based on today’s em-
pirical knowledge on the breaking of random waves (Banner et al. 2000; Babanin et al. 2001) and the dissipation
of swells over long distances (Ardhuin et al. 2009b). The
present formulations are still semi-empirical, in the sense
that they are not based on a detailed physical model of dissipation processes, but they demonstrate that progress is
possible. This effort opens the way for completely physical
parameterizations (e.g. Filipot et al. 2010) that will eventually provide new applications for wave models, such as
the estimation of statistical parameters for breaking waves,
including whitecap coverage and foam thickness. Other efforts, less empirical in nature, are also under way to arrive
at better parameterizations (e.g. Banner and Morison 2006;
Babanin et al. 2007; Tsagareli 2008), but they yet have to
produce a practical alternative for wave forecasting and
hindcasting.
b. Shortcomings of existing parameterizations
All wave dissipation parameterizations up to the work
of van der Westhuysen et al. (2007) had no quantitative relationship with observed features of wave dissipation, and
the parameterizations were generally used as set of tuning
knobs to close the wave energy balance. The parameterization of the form proposed by Komen et al. (1984) have
produced a family loosely justified by the so-called ‘random
pulse’ theory of (Hasselmann 1974). These take a generic
form
"
2 #
k
k
0.5 4.5 4
,
(2)
Soc (f, θ) = Cds g kr Hs δ1 + δ2
kr
kr
in which Cds is a negative constant, and kr is an energyweighted mean wavenumber defined from the entire spectrum, and Hs is the significant wave height. In the early
and latest parameterizations, the following definition was
used
"
#1/r
Z
Z
16 fmax 2π r
kr =
k E (f, θ) df dθ
,
(3)
Hs2 0
0
where r is a chosen real constant, typically r = −0.5 or
r = 0.5.
These parameterizations are still widely used in spite of
inconsistencies in the underlying theory. Indeed, if whitecaps do act as random pressure pulses, their average work
on the underlying waves only occurs because of a phase correlation between the vertical orbital velocity field and the
moving whitecap position, which travels with the breaking
wave. In reality the horizontal shear is likely the dominant
mechanism (Longuet-Higgins and Turner 1974), but the
question of correlation remains the same. For any given
whitecap, such a correlation cannot exist for all spectral
wave components: a whitecap that travels with one wave
leads to the dissipation of spectral wave components that
1 In the presence of variable current, the source of energy for the
wave field, i.e. the work of the radiation stresses, is generally hidden
when the energy balance is written as an action balance (e.g. Komen
et al. 1994).
2
propagate in similar directions, with comparable phase velocities. However, whitecaps moving in one direction will
give (on average) a zero correlation for waves propagating
in the opposite direction because the position of the crests
of these opposing waves are completely random with respect to the whitecap position. As a result, not all wave
components are dissipated by a given whitecap (others
should even be generated), and the dissipation function
cannot take the spectral form given by Komen et al. (1984).
A strict interpretation of the pressure pulse model gives
a zero dissipation for swells in the open ocean because the
swell wave phases are uncorrelated to those of the shorter
breaking waves. There is only a negligible dissipation due
to short wave modulations by swells and preferential breaking on the swell crests (Phillips 1963; Hasselmann 1971;
Ardhuin and Jenkins 2005). Still, the Komen et al. (1984)
type dissipation terms are applied to the entire spectrum,
including swells, without any physical justification.
In spite of its successful use for the estimation of the
significant wave height Hs and peak period Tp , these fixedshape dissipation functions, from Komen et al. (1984) up
to Bidlot et al. (2007a), have built-in defects. Most conspicuous is the spurious amplification of wind sea growth
in the presence of swell (e.g. van Vledder and Hurdle 2002),
which is contrary to all observations (Dobson et al. 1989;
Violante-Carvalho et al. 2004; Ardhuin et al. 2007). Associated wih that defect also comes an underestimation of
the energy level in the inertial range, making these wave
models ill-suited for remote sensing applications, as will be
exposed below.
Also, these parameterizations typically give a decreasing dissipation of swell with increasing swell steepness, contrary to all observations from Darbyshire (1958) to Ardhuin
et al. (2009b). This effect is easily seen by taking a sea state
composed of a swell and wind sea of energy E1 and E2 and
mean wavenumbers k1 and k2 , respectively, with k2 > k1 .
The overall mean wavenumber is
kr = [(k1r E1 + k2r E2 ) /(E1 + E2 )]1/r .
An alternative and widely used formulation has been
proposed by Tolman and Chalikov (1996), and some of its
features are worth noting. It combines two distinct dissipation formulations for high and low frequencies, with a transition at two times the wind sea peak frequency. Whereas
Janssen et al. (1994) introduced the use of two terms, k and
k 2 in eq. (2), in order to match the very different balances
in high and low frequency parts of the spectrum, they still
had a common fixed coefficient, Cds g 0.5 kr4.5 Hs4 . In Tolman
and Chalikov (1996) these two dissipation terms are completely distinct, the low frequency part being linear in the
spectrum and proportional to wind friction velocity u⋆ , the
high frequency part is also linear and proportional to u2⋆ . In
this formulation the frequency dependence of the two terms
is also prescribed. Tolman and Chalikov (1996) further included swell attenuation by the wind, based on numerical
simulations of the airflow above waves (Chalikov and Belevich 1993), here noted Sout . At relatively short fetches,
these source terms are typically a factor 2 to 3 smaller than
those of Janssen et al. (1994), which was found to produce
important biases in wave growth and wave directions at
short fetch (Ardhuin et al. 2007). Another set of parameterizations was proposed by Makin and Stam (2003). It
is appropriate for high winds conditions but does not produce accurate results in moderate sea states (Lefèvre et al.
2004). Finally, among the many formulations proposed we
may cite one by Polnikov and Inocentini (2008), but its
accuracy appears generally less than with the model presented here, in particular for mean periods.
Based on observations of large wave height gradients in
rapidly varying currents, Phillips (1984) proposed a dissipation rate proportional to the non-dimensional spectrum
B, also termed ’saturation spectrum’. Banner et al. (2000)
indeed found a correlation between the breaking probability of dominant waves and the saturation, when the latter is
integrated over a finite frequency bandwidth and all directions., with breaking occuring when B exceeds a threshold
Br . Alves and Banner (2003) proposed to define the dissipation Soc by B/Br to some power, multiplied by a Komentype dissipation term. Although this approach avoided the
investigation of the dissipation of non-breaking waves, it
imported all the above mentionned defects of that parameterization. Further, these authors used a value for Br that
is much higher than suggested by observations, which tends
to disconnect the parameterization from the observed effects (Babanin and van der Westhuysen 2008).
The use of a saturation parameter was taken up again
by van der Westhuysen et al. (2007), hereinafter WZB,
who, Like Alves and Banner (2003), integrated the saturation spectrum over directions, giving
Z 2π
B (f ) =
k 3 F (f, θ′ )Cg /(2π)dθ′ .
(5)
(4)
Equation (2) gives a dissipation that is proportional to
kr3.5 (E1 + E2 ) in the low frequency limit. Now, if we keep
k1 , k2 an E2 constant and only increase the swell energy E1 ,
the relative change in dissipation is, according to (2), proportional to x = 3.5[(k1 /kr )r −1]/r+2. For r = 0.5, as used
by (Bidlot et al. 2005, hereinafter BAJ), and k1 /kr < 0.51,
x is negative (i.e. the dissipation decreases with increasing
swell energy). For equal energy in sea and swell, this occurs
when k1 /k2 < 0.3, which is generally the case with sea and
swell in the ocean. This erroneous decrease of swell dissipation with increasing swell steepness is reduced when the
model frequency range is limited to maximum frequency of
0.4 Hz, in which case the lowest winds (less than 5 m/s) are
unable to produce a realistic wind sea level, hence limiting
the value of kr to relatively small values.
0
3
From this, they defined the source function
p/2
p
B(f )
Soc,WZB (f, θ) = −C gk
,
Br
rameterizations at all scales, in order to provide a robust
and comprehensive parameterization of wave dissipation.
(6)
c. A new set of parameterizations and adjustments to get adequate balances
where C is a positive constant, Br is a constant saturation
threshold and and p is a coefficient that varies both with
the wind friction velocity u⋆ and the degree of saturation
B(f )/Br with, in particular, p ≈ 0 for B(f ) < 0.8Br . For
non breaking waves, when p ≈ 0, the dissipation is too large
by at least one order of magnitude, making the parameterization unfit for oceanic scale applications with wave heights
in the Atlantic underpredicted by about 50% (Ardhuin and
Le Boyer 2006). In van der Westhuysen (2007) this was
addressed by reverting back to Komen et al. (1984) dissipation for non-breaking waves, but no solution for the
dissipation of these spectral components was proposed. In
WZB, the increase of p with the inverse wave age u⋆ /C
was designed to increase Soc at high frequency, which was
needed to obtain a balance with the Satm term in equation
(1). This indicates that, besides the value of the saturation
Br , other factors may be important, such as the directionality of the waves (Banner et al. 2002). Other observations
clearly show that the breaking rate of high frequency waves
is much higher for a given value of B, probably due to cumulative effects by which the longer waves are modifying
the dissipation of shorter waves.
Banner et al. (1989) and Melville et al. (2002) have
shown how breaking waves suppress the short waves on the
surface, and we will show here that a simple estimation of
the dominant breaking rates based on the observations by
Banner et al. (2000) suggests that this effect is dominant
for wave frequencies above three times the windsea peak
frequency. Young and Babanin (2006) arrived at the same
conclusion from the examination of wave spectra, and proposed a parameterization for Soc that included a new term,
the cumulative term, to represent theis effect. Yet, their
estimate was derived for very strong wind-forcing conditions only. Further, their interpretation of the differences
in parts of a wave record with breaking and non-breaking
waves implies an underestimation of the dissipation rates
because the breaking waves have already lost some energy
when they are observed and the non-breaking waves are not
going to break right after they have been observed. Also,
since the spectra are different, nonlinear interactions must
be different, even on this relatively small time scale (e.g.
Young and van Vledder 1993, figure 5), and the differences
in spectra may not be the result of dissipation alone.
Finally, the recent measurement of swell dissipation by
Ardhuin et al. (2009a) has revealed that the dissipation of
non-breaking waves is essentially a function of the wave
steepness, and a very important process for ocean basins
larger than 1000 km. Because of the differences between
coastal and larger scale sea states (e.g. Long and Resio
2007), it is paramount to verify the source function pa-
It is thus time to combine the existing knowledge on the
dissipation of breaking and non-breaking waves to provide
an improved parameterization for the dissipation of waves.
Our objective is to provide a robust parameterization that
improves existing wave models. For this we will use the
parameterization by BAJ as a benchmark, because it was
shown to provide the best forecasts on global scales (Bidlot
et al. 2007b) before the advent of the parameterizations
presented here. BAJ is also fairly close to the widely used
“WAM-Cycle 4” parameterization by Janssen and others
(Komen et al. 1994).
We will first present a general form of the dissipation
terms based on observed wave dissipation features. The
degrees of freedom in the parameterization are then used
to adjust the model result. In particular we adujst a cumulative breaking effect and a wind sheltering effect that,
respectively, dissipates and reduces the wind input to short
waves as a function of longer waves characteristics. A comprehensive validation of wave parameters is then presented
using field experiments and a one year hindcast of waves
at the global and regional scale, in which all possible wave
measurements are considered, with significant wave heights
ranging from 0 to 17 m. The model is further validated
with independent data at regional and global scales.
Tests and verification in the presence of currents, and
using a more realistic parameterizations of wave-wave interactions will be presented in parts II and III. These may
also include some replacement of the arbitrary choices made
here in the details of the dissipation parameters, with physicallymotivated expressions.
2. Parameterizations
Several results will be presented, obtained by a numerical integration of the energy balance. Because numerical choices can have important effects (e.g. Tolman
1992; Hargreaves and Annan 2000), a few details should
be given. All calculations are performed with the WAVEWATCH IIITM modelling framework (Tolman 2008, 2009),
hereinafter WWATCH, using the third order spatial and
spectral advection scheme, and including modifications of
the source terms described here. In all cases ran with
WWATCH, the source terms are integrated with the fully
implicit scheme of Hargreaves and Annan (2000), combined
with the adaptative time step and limiter method of Tolman (2002a), in which a minimum time step of 10 s is
used, so that the limiter on wind-wave growth is almost
never activated. The diagnostic tail, proportional to f −5
4
A few tests have indicated that a threshold Rec = 2 ×
105 m/Hs provides reasonable result, although it may be
a also be a function of the wind speed, and we have no
explanation for the dependence on Hs . A constant threshold close to 2 × 105 provides similar results. Here we shall
use Cdsv = 1.2, but the results are not too sensitive to the
exact value.
The parameterization of the turbulent boundary layer
is more problematic. Without direct measurements in the
boundary layer, there is ample room for speculations. From
the analogy with an oscillatory boundary layer over a fixed
bottom (Jensen et al. 1989), the values of fe inferred from
the swell observations, in the range 0.004 to 0.013 (Ardhuin
et al. 2009b), correspond to a surface with a very small
roughness. Because, we also expect the wind to influence
fe , the parameterization form includes adjustable effects
of wind speed on the roughness, and an explicit correction
of fe . This latter correction takes the form of a Taylor
expansion to first order in u⋆ /uorb ,
u⋆
fe = s1 fe,GM + [|s3 | + s2 cos(θ − θu )]
,
(10)
uorb
is only imposed at a cut-off frequency fc set to
fc = fFM fm .
(7)
Here we take fFM = 10 and define the mean frequency as
fm =1/Tm0,1. Hence fc is generally above the maximum
model frequency that we fixed at 0.72 Hz, and the high
frequency tail is left to evolve freely. Some comparison tests
are also done with other parameterizations using a lower
value of fc , typically set at 2.5 fm (Bidlot et al. 2007a).
In such calculations, although the net source term may
be non-zero at frequencies above fc , there is no spectral
evolution due to the imposed tail.
a. Nonlinear wave wave interactions
All the results discussed and presented in this section
are obtained with the Discrete Interaction Approximation
of Hasselmann et al. (1985). The coupling coefficient that
gives the magnitude of the interactions is Cnl . Based on
comparisons with exact calculations, Komen et al. (1984)
adjusted the value of Cnl to 2.78 × 107 , which is the value
used by Bidlot et al. (2005). Here this constant will be allowed to vary slightly. This parameterization is well known
for its shortcomings (Banner and Young 1994), and the
adjustment of other parameters probably compensates for
some of these errors. This matter will be fully discussed in
Part III.
where fe,GM is the friction factor given by Grant and Madsen’s (1979) theory for rough oscillatory boundary layers
without a mean flow. Adequate swell dissipation is obtained with constant values of fe in the range 0.004 to
0.007, but these do not necessarily produce the best results when comparing wave heights to observations. Based
on the simple idea that most of the air-sea momentum flux
is supported by the pressure-slope correlations that give
rise to the wave field (Donelan 1998; Peirson and Banner
2003), we have set the surface roughness to
b. Swell dissipation
Observations of swell dissipation are consistent with the
effect of friction at the air-sea interface (Ardhuin et al.
2009a), resulting in a flux of momentum from the wave
field to the wind (Harris 1966). We thus write the swell
dissipation as a negative contribution Sout which is added
to Sin to make the wind-wave source term Satm .
Using the method of Collard et al. (2009), a systematic
analysis of swell observations by Ardhuin et al. (2009a)
showed that the swell dissipation is non-linear, possibly
related to a laminar-to-turbulent transition of the oscillatory boundary layer over swells. Defining the boundary
Reynolds number Re= 4uorb aorb /νa , where uorb and aorb
are the significant surface orbital velocity and displacement
amplitudes, and νa is the air viscosity, we take, for Re less
than a critical value Rec
o
ρa n √
2k 2νσ F (f, θ) ,
(8)
Sout (f, θ) = −Cdsv
ρw
z0′ = rz0 z0
where rz0 is here set to 0.04, of the roughness for the wind.
We thus give the more generic equation (10) for fe , with
fe,GM of the order of 0.003 for values of aorb /z0′ of the order
of 2×105.
The coefficients s2 and s3 of the O(u⋆ /uorb ) correction have been adjusted to -0.018 and 0.015, respectively,
the former negative value giving a stronger dissipation for
swells opposed to winds, when cos(θ − θu ) < 0. This gives
a range of values of fe consistent with the observations,
and reasonable hindcasts of swell decay (Fig. 1), with a
small underestimation of dissipation for steep swells. An
increase of s1 from 0.8 to 1.1 produces negative biases on
Hs of the order of 30% at all oceanic buoys (40% for a partial wave height estimated from a spectrum restricted to
periods around 15 s), so that the magnitude of the swell dissipation cannot be much larger than chosen here. Further
discussion and validation of the swell dissipation is provided by the global scale hindcasts in Section 4.
where the constant Cdsv is equal to 1 in Dore (1978)’s laminar theory.
When the boundary layer is expected to be turbulent,
for Re≥Rec , we take
Sout (f, θ) = −
ρa 16fe σ 2 uorb /g F (f, θ) .
ρw
(11)
(9)
5
estimating a breaking probability.
We started from the simplest possible dissipation term
formulated in terms of the direction-integrated spectral saturation B (f ) given by eq. (5), with a realistic threshold
B0r = 1.2×10−3 corresponding to the onset of wave breaking (Babanin and Young 2005). This saturation parameter
corresponds exactly to the α parameter defined by Phillips
(1958). The value B0 = 8 × 10−3 , given by Phillips, corresponds to a self-similar sea state in which waves of all
scales have the same shape, limited by the breaking limit.
This view of the sea state, however, ignores completely
wave directionality. Early tests of parameterizations based
on this definition of B indicated that the spectra were too
narrow (Ardhuin and Le Boyer 2006). This effect could
be due to many errors. Because Banner et al. (2002) introduced a directional width in their saturation to explain
some of the variability in observed breaking probabilities,
we similarly modify the definition of B. Expecting also
to have different dissipation rates in different directions,
we define a saturation B ′ that would correspond, in deep
water, to a normalized velocity variance projected in one
direction (in the case sB = 2), with a further restriction of
the integration of directions controlled by ∆θ ,
Fig. 1. Comparison of modelled swell significant heights,
following the propagation of the two swells shown by Ardhuin et al. (2009) with peak periods of 15 s and high and
low dissipation rates. Known biases in the level 2 data have
been corrected following Collard et al. (2009).
B ′ (f, θ) =
Z
θ+∆θ
θ−∆θ
k 3 cossB (θ − θ′ ) F (f, θ′ )
Cg ′
dθ ,
2π
(12)
Here we shall always use ∆θ = 80◦ . As a result, a sea state
with two systems of same energy but opposite direction will
typically produce much less dissipation than a sea state
with all the energy radiated in the same direction.
We finally define our dissipation term as the sum of
the saturation-based term of Ardhuin et al. (2008a) and a
cumulative breaking term Sbk,cu,
h
C sat
Soc (f, θ)
= σ Bds2 δd max {B (f ) − Br , 0}2
r
i
2
+ (1 − δd ) max {B ′ (f, θ) − Br , 0} F (f, θ)
c. Wave breaking
Observations show that waves break when the orbital
velocity at their crest Uc comes close to the phase speed
C, with a ratio Uc /C > 0.8 for random waves (Tulin and
Landrini 2001; Stansell and MacFarlane 2002; Wu and Nepf
2002). It is nevertheless difficult to parameterize the breaking of random waves, since the only available quantity here
is the spectral density. This density can be related to the
orbital velocity variance in a narrow frequency band. This
question is addressed in detail by Filipot et al. (2010). Yet,
a proper threshold has to be defined for this quantity, and
the spectral rate of energy loss associated to breaking has
to be defined. Also, breaking is intricately related to the
complex non-linear evolution of the waves (e.g. Banner and
Peirson 2007).
These difficulties will be ignored here. We shall parameterize the spectral dissipation rate directly from the wave
spectrum, in a way similar to WZB. Essentially we distiguish between spontaneous and induced breaking, the latter being caused by large scale breakers overtaking shorter
waves, and causing them to be dissipated. For the spontaneous breaking we parameterize the dissipation rate directly from the spectrum, without the intermediate step of
+
Sbk,cu (f, θ) + Sturb (f, θ).
(13)
where
B (f ) = max {B ′ (f, θ), θ ∈ [0, 2π[} .
(14)
The combination of an isotropic part (the term that
multiplies δd ) and a direction-dependent part (the term
with 1 − δd ) was intended to allow some control of the
directional spread in resulting spectra. This aspect is illustrated in figure 2 with a hindcast of the November 3 1999
case during the Shoaling Waves Experiment (Ardhuin et al.
2007). Clearly, the isotropic saturation in the TEST442
dissipation (with the original threshold Br = 0.0012) produces very narrow spectra, even though it is known that
the DIA parameterization for nonlinear interactions, used
here, tends to broaden the spectra. The same behaviour
is obtained with the isotropic parameterization by van der
6
(a)
f (Hz)
(b)
Ccu=−1, su=0
Ccu=−0.4, su=1
2 Ccu=−0.4, su=1
= 1, Br = 0.0012
3 Ccu=−0.4, su=1
= 0, Br = 0.0009
(c)
Fig. 2. Wave spectra on 3 November 1999 at buoy X3
(fetch 39 km, wind speed U10 = 9.4 m s−1 ), averaged over
the time window 1200-1700 EST, from observations and
model runs, with different model parameterizations (symbols): BAJ stands for Bidlot et al. (2005). (a) Energy, (b)
mean direction (c) directional spread. This figure is analogue to the figures 10 and 11 in Ardhuin et al. (2007), the
model forcing and setting are identical. It was further verified that halving the resolution from 1 km to 500 m does
not affect the results. All parameters for BAJ, TEST441
and TEST443 are listed in tables 3 and 4. Input parameters for TEST443 are identical to those for TEST441, and
TEST442 differs from TEST441 only in its isotropic direct breaking term, given by sB = 0, ∆θ = 180◦, and
Br = 0.0012. It should be noted that the overall dissipation term in TEST443 is made anisotropic due to the
cumulative effect, but this does not alter much the underestimation of directional spread.
Fig. 3. Fetch-limited growth of the windsea energy as
a function of fetch on 3 November 1999, averaged over
the time window 1200-1700 EST, from observations and
model runs, with different model parameterizations (symbols): BAJ stands for Bidlot et al. (2005). This figure is
analogue to the figure 8 in Ardhuin et al. (2007), the model
forcing and setting are identical. All parameters for BAJ,
TEST441 and TEST443 are listed in tables A1 and A2.
Input parameters for TEST443 are identical to those for
TEST441, and TEST442 differs from TEST441 only in its
isotropic direct breaking term, given by sB = 0, ∆θ = 180◦ ,
and Br = 0.0012.
possible levels of input. Here the mean directions at the
observed peak frequency are still biassed by about 25◦ towards the alongshore direction with the parameterizations
proposed here (Fig. 2.b), which is still less that the 50◦ obtained with the weaker Tolman and Chalikov (1996) source
terms (Ardhuin et al. 2007, figure 11). A relatively better
fit is obtained with the BAJ parameterization. This is
likely due to either the stronger wind input or the weaker
dissipation at the peak. It is likely that both features of
the BAJ parameterization are more realistic than what we
propose here.
Figure 3 shows the fetch-limited growth in wave energy
of various parameterizations. We repeat here the sensitivity test to the presence of swell, already displayed in
Ardhuin et al. (2007). Whereas the 1 m swell causes an
unrealistic doubling of the wind sea energy at short fetch
in the BAJ parameterization, the new parameterizations,
just like the one by van der Westhuysen et al. (2007) are,
by design, insensitive to swell (not shown).
The dissipation Sturb due to wave-turbulence interac-
Westhuysen et al. (2007), as demonstrated by Ardhuin and
Le Boyer (2006). Further, using an isotropic dissipation at
all frequencies yields an energy spectrum that decays faster
towards high frequencies than the observed spectrum (Fig.
2.a). On the contrary, a fully directional dissipation term
(TEST443 with δd = 0) gives a better fit for all parameters.
With sB = 2, we reduce Br to 0.0009, a threshold for the
onset of breaking that is consistent with the observations
of Banner et al. (2000) and Banner et al. (2002).
sat
The dissipation constant Cds
was adjusted to 2.2 ×
−4
10 in order to give acceptable time-limited wave growth
and reasonable directions in fetch-limited growth (Ardhuin et al. 2007). As noted in this previous work, similar
growths of wave energy with fetch are possible with almost
any magnitude of the wind input, but a reasonable mean
direction in slanting fetch conditions selects the range of
7
However, because they used a zero-crossing analysis, for a
given wave scale, there are many times when waves are not
counted because the record is dominated by another scale:
in their analysis there is only one wave at any given time.
This tends to overestimate the breaking probability by a
factor of 1.5 to 2 (Manasseh et al. 2006), compared to the
present approach in which we consider that several waves
(of different scales) may be present at the same place and
time. We shall thus correct for this effect, simply dividing
P by 2.
With this approach we define the spectral density of
crest Rlength (breaking or not) per unit surface l(k) such
that l(k)dkx dky is the total length of all crests per unit
surface, with a crest being defined as a local maximum of
the elevation in one horizontal direction. In the wavenumber vector spectral space we take
tions (Ardhuin and Jenkins 2006) is expected to be much
weaker than all other terms and will be neglected here.
Finally, following the analysis by Filipot et al. (2010),
the threshold Br is corrected for shallow water, so that
B ′ /Br in different water depths corresponds to the same
ratio of the root mean square orbital velocity and phase
speed. For periodic and irrotational waves, the orbital velocity increases much more rapidly than the wave height as
it approaches the breaking limit. Further, due to nonlinear
distortions in the wave profile in shallow water, the height
can be twice as large as the height of linear waves with the
same energy. In order to express a relevant threshold from
the elevation variance, we consider the slope kHlin (kD) of
an hypothetical linear wave that has the same energy as
the wave of maximum height. In deep water2 , kHlin (∞) ≈
0.77, and for other water depths we thus correct Br by a
factor (kHlin (kD)/Hlin (∞))2 . Using streamfunction theory (Dalrymple 1974), a polynomial fit as a function of
Y = tanh(kD) gives
(15)
Br′ = Br Y M4 Y 3 + M3 Y 2 + M2 Y + M1 .
l(k) = 1/(2π 2 k)
(17)
which is equivalent to a constant in wavenumber-direction
space l(k, θ) = 1/(2π 2 ). This number was obtained by
considering an ocean surface full of unidirectional waves,
such that Br′ = Br in deep water. The fitted constants
with one crest for each wavelength 2π/k for each spectral
are M4 = 1.3286, M3 = −2.5709 , M2 = 1.9995 and
interval ∆k = k, e.g. one crest corresponding to spectral
M1 = 0.2428. Although this behaviour is consistent with
components in the range 0.5 k to 1.5 k. We further double
the variation of the depth-limited breaking parameter γ
the potential number of crests to account for the directionderived empirically by Ruessink et al. (2003), the resultality of the sea state. These two assumptions have not
ing dissipation rate is not yet expected to produce realistic
been verified and thus the resulting value of l(k) is merely
results for surf zones because no effort was made to verify
an adjustable order of magnitude.
this aspect. This is the topic of ongoing work, outside of
Thus the spectral density of breaking crest length per
the scope of the present paper.
unit surface is Λ(k) = l(k)P (k). Assuming that any breakThe cumulative breaking term Sbk,cu represents the smoothing wave instantly dissipates all the energy of all waves
ing of the surface by big breakers with celerity C ′ that wipe
with frequencies higher by a factor rcu or more, then the
out smaller waves of phase speed C. Due to uncertainties
cumulative dissipation rate is simply given by the rate at
in the estimation of this effect in the observations of Young
which these shorter waves are taken over by larger breaking
and Babanin (2006), we use the theoretical model of Ardwaves, times the spectral density, namely
huin et al. (2009b). Briefly, the relative velocity of the
Z
crests is the norm of the vector difference, ∆C = |C − C′ |,
∆C Λ(k′ )dk′ ,
(18)
S
(f,
θ)
=
C
F
(f,
θ)
bk,cu
cu
and the dissipation rate of short wave is simply the rate of
f ′ <rcu f
passage of the large breaker over short waves, i.e. the integral of ∆C Λ(C)dC, where Λ(C)dC is the length of breaking crests per unit surface that have velocity components
between Cx and Cx + dCx , and between Cy and Cy + dCy
(Phillips 1985). Because there is no consensus on the form
of Λ (Gemmrich et al. 2008), we prefer to link Λ to breaking probabilities. Based on Banner et al. (2000, figure
6, bT = 22 (ε − 0.055)2 ), and taking
p their saturation parameter ε to be of the order of 1.6 B ′ (f, θ), the breaking
probability of dominant waves waves is approximately
2
p
p
P = 56.8 max{ B ′ (f, θ) − Br′ , 0} .
(16)
where rcu defines the maximum ratio of the frequencies of
long waves that will wipe out short waves.
We now obtain Λ by extrapolating eq. (16) to higher
frequencies,
R r f R 2π
Ccu F (f, θ) 0 cu 0 28.4
π
np
o2
√
′
′
C
× max
B(f ′ , θ′ ) − Br , 0 ∆
C ′ dθ df ,
Sbk,cu(f, θ) =
g
(19)
We shall take rcu = 0.5, and Ccu is a tuning coefficient
expected to be a negative number of order 1, which also
corrects for errors in the estimation of l.
This generic form of the source terms produces markedly
different balances for both mature and fully developped
2
√ This value of the maximum equivalent linear height Hlin =
2 2E, with E the elevation variance, is smaller than the usual value
kH = 0.88 √
due to the correction for the nonlinear wave profile for
which H > 2E.
8
The most important qualitative feature is the lack of
a regular predefined shape for the normalized dissipation
term Soc (f )/E(f ). Whereas the shape given by δk + (1 −
δ)k 2 is clearly visible in BAJ (with extremely high dissipation rates if one considers high frequencies), and the low to
high frequency dissipation transition at 2 fp is evident in
TC, the shape of the new dissipation rates are completely
dictated by the local spectral saturation level. This leads
to a relatively narrow peak of dissipation right above the
spectral peak, where saturation is strongest.
This feature helps to produce the realistic spectral shapes
near the peak, with a steeper low frequency side and a more
gentle slope on the high frequency side, contrary to the
backward facing spectra produced by BAJ and TC. However, this localized strong relative dissipation, Soc (f )/E(f ),
is hard to reconcile with time and spatial scales of breaking events, and thus probably exaggerated. Indeed, there
should be no significant difference in relative dissipation
among the spectral components that contribute to a breaking wave crest, provided that they do not disperse significantly over the breaker life time, which is less than a wave
period. Linear dispersion of waves with frequencies that
differ by only 10% should only produce small relative phase
shift. Thus, there is no physical reason why a breaking
event would take much more energy, relatively speaking,
from the spectral band 1.1 to 1.2fp than from 1.2 to 1.3fp .
The factor 2 difference produced here in the relative dissipation rates is unrealistic. This strong relative dissipation
at the peak (50% higher than in BAJ) is one important
factor that gives a slower growth of the wave spectrum
in TEST441 compared to BAJ. It is possible that the localization of Soc at the peak compensates for the broader
spectrum produced by the DIA compared to results with
an exact non-linear interaction calculation.
We now consider “fully developped” conditions, illustrated by figure 5, corresponding to long durations with
steady wind and infinite fetch. At low frequency, the nonlinear swell damping term Sout (the negative part of Sin )
cancels about 30 to 50% of the nonlinear energy flux, so
that the sea state grows only very slowly. As a result
“full development” does not exist (the wave height keeps
growing), but the resulting energy is still compatible with
the observations of mature sea states (Alves et al. 2003).
In contrast, the linear swell damping adjusted by Tolman
(2002b) to produce reasonable swell heights in the tropics is much smaller than the non-linear energy flux to low
frequencies, even with the reduced interaction coefficient
proposed by Tolman and Chalikov (1996). A non-linear
swell dissipation appears necessary to obtain both a realistic damping of observed swells and a satisfatory agreement
with mature wind waves. Nonlinearity also bring within
the same order of magnitude the decay scales estimated
for short (Hogstrom et al. 2009) and very long swells (Ardhuin et al. 2009b).
spectra E(f) and normalized source terms M(f)* S(f)
0.2
0.1
0
−0.1
BAJ
−0.2
0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.1
0
−0.1
−0.2
0
Frequency (Hz)
0.2
0.1
0
−0.1
437
=−1
−0.2
0
0.2
0.4
0.6
0.8
0.4
0.6
0.8
0.4
0.6
0.8
0.2
0.1
0
−0.1
−0.2
=−0.4
0
0.2
0.2
E(f)
0.1
Satm
Snl
Soc
Stot
0
−0.1
−0.2
0
441
T&C
0.2
Fig. 4. Academic test case over a uniform ocean with a
uniform 10 m s−1 wind starting from rest, after 8 hours
of integration, when Cp /U10 ≈ 1. Source term balances
given by the parameterization BAJ, and the parameterizations proposed here with the successive introduction
of the cumulative breaking and the wind sheltering effects with the parameters Ccu and su . For BAJ, a diagnostic f −5 tail is applied above 2.5 the mean frequency.
In order to make the high frequency balance visible, the
source terms are multiplied by the normalization function
M (f ) = ρw C/(ρa E(f )σU10 ). The result with the parameterization of Tolman and Chalikov (1996) is also given for
reference.
seas. For mature seas, without cumulative effect, figure
(4) shows that a balance is possible that gives roughly the
same energy level and wind input term as the BAJ parameterization, up to 0.4 Hz. However, the balance for higher
frquencies produces energy levels decreasing slower than
f −4 as the dissipation is too weak compared to the input,
and thus the nonlinear energy flux is reversed, pumping
energy from the tail to lower frequencies.
The introduction of a strong cumulative term (TEST437)
allows a balance at roughly the same energy level. However, with the present formulation this will lead to a dissipation too strong at high frequency for higher winds. The
introduction of the sheltering effect via the parameter su
(details in section d) is designed to get a balance with a
weaker cumulative effect.
9
0.012
0.01
B0
0.008
BAJ
BAJ no tail
Ccu=0,su=0
Ccu=1., su=0
Ccu=0.4, su=1
TC
Phillips
(1958)
0.006
0.004
0.002
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency (Hz)
Fig. 6. Values of the spectral saturation B0 for the cases
presented in figure 5.
=−1
=−0.4
Fig. 5. Same as figure (4) but after 48 h of integration
and without normalization of the source terms. Source
term balances given by the parameterization BAJ, and the
parameterizations proposed here with the successive introduction of the cumulative breaking and the wind sheltering
effects with the parameters Ccu and su . For BAJ, a diagnostic f −5 tail is applied above 2.5 the mean frequency.
Both parameterizations are physically very different from
the parameterizations of the Komen et al. (1984) family, including Bidlot et al. (2005). In these, the swell energy is
lost to the ocean via whitecapping. Here we propose that
this energy is lost to the atmosphere, with an associated
momentum flux that drives the wave-driven wind observed
in laboratories (Harris 1966) and for very weak winds at
sea (Smedman et al. 2009).
In the inertial range, a reasonable balance of all the
source terms is obtained for Ccu = −0.4 (figure 5). In this
case, the spectrum approaches an f −4 shape, for which Snl
goes to zero, whereas in BAJ it decreases even faster than
f −5 which makes Snl positive above 0.4 Hz. The behaviour
of the high frequency tail is best seen when displayed in
non-dimensional form, as done in figure 6. This shows the
unrealistic high level of the tail without cumulative effect
10
nor modification of the wind input, and the equally unrealistic low tail with the BAJ parameterization, especially
when the tail is left to evolve freely.
Adding the cumulative effect can be used to control
the tail level, but this degree of freedom is not enough.
Indeed, in strongly forced conditions the dominant waves
break frequently, and a high cumulative effect, Ccu = −1,
reduces the energy level in the tail below observed levels. This effect can be seen by considering satellite-derived
mean square slopes (Fig. 8), or high moments of the frequency spectrum derived from buoy data (not shown but
similar).
That effect can be mitigated by decreasing Ccu or increasing rcu , so that dominant breaking waves will only
wipe out much smaller waves. Instead, and because the
wind to wave momentum flux was apparently too high in
high winds, we chose to introduce one more degree of freedom, allowing a reduction of the wind input at high frequency.
d. Wind input
The wind input parameterization is thus adapted from
Janssen (1991) and the following adjustments performed
by Bidlot et al. (2005, 2007a). The full wind input source
term reads
ρa βmax Z 4 u⋆ 2
up
e Z
Sin (f, θ) = Sin
(f, θ) +
ρw κ 2
C
p
× max {cos(θ − θu ), 0} σF (f, θ) , (20)
where βmax is a non-dimensional growth parameter (constant), κ is von Kármán’s constant. In the present implementation the air/water density ratio is constant. The
power of the cosine is taken constant with p = 2. We define the effective wave age Z = log(µ) where µ is given
1
Z = log(kz1 ) + κ/ [cos (θ − θu ) (u⋆ /C + zα )] ,
0.15 E(f) or 10000 S(f)
by Janssen (1991), and corrected for intermediate water
depths, so that
(21)
0.5
where z1 is a roughness length modified by the wave-supported
0
E(f)
stress τw , and zα is a wave age tuning parameter. z1 is imS BAJ
in
plicitly defined by
+z =0.006
a
+
=1.52
22
0.5
zu
u⋆
+s =1
u
(22)
log
U10 =
+S
κ
z1
out
τ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
z0 = min α0 , z0,max
(23)
Frequency (Hz)
g
z0
z1 = p
,
(24)
Fig. 7. Incremental adjustements to the wind-wave in1 − τw /τ
teraction source term Satm , going from the BAJ form to
where zu is the height at which the wind speed is specthe one used in TEST441. From one curve to the next,
ified, usually 10 meters. The maximum value of z0 was
only one parameter is changed. The computations are peradded to reduce the unrealistic wind stresses at high winds
formed for the same spectrum obtained by running the
that are otherwise given by the standard parameterizamodel from a calm sea for 8 hours with the BAJ paramtion. For example, z0,max = 0.0015 is equivalent to setting
eterization and a wind speed of 10 m s−1 . The reduction
a maximum wind drag coefficient of 2.5 × 10−3 . For the
of zα from 0.011 to 0.006 strongly reduces the input for
TEST441 parameterization, we have adjusted zα = 0.006
frequencies in the range 0.15 to 0.2 Hz, which is probably
and βmax = 1.52 (Fig. 7).
overestimated in BAJ when average (20%) levels of gustiAn important part of the parameterization is the calness are considered: now the wind input goes to zero for
culation of the wave-supported stress τw , which includes
f = 0.13, which corresponds to C/U10 = 0.83, whereas
the resolved part of the spectrum, as well as the growth
it is still significant at that wave age in the BAJ parameof an assumed f −5 diagnostic tail beyond the highest freterization. As a result the much lower input level need a
quency. This parameterization is highly sensitive to the
readjustment, performed here by increasing βmax to 1.52.
high frequency part of the spectrum since a high energy
Yet, this high value of βmax produces very high wind stress
level there will lead to a larger value of z1 and u⋆ , which
values and thus a very strong high frequency input. Adding
gives a positive feedback and reinforces the energy levels.
the sheltering term su = 1 allows a decent balance at high
In order to allow a balance with the saturation-based
frequency. Finally, the addition of the air-sea friction term
dissipation, the wind input at high frequency is reduced by
that gives swell dissipation produces a significant reduction
modifying the friction velocity u⋆ . This correction also
of the input to the wind sea at f = 0.25 Hz. It is questionreduces the drag coefficient at high winds. Essentially,
able whether this mechanism also applies in the presence
the wind input is reduced for high frequencies and high
of the critical layer for those waves. This matter clearly
winds, loosely following Chen and Belcher (2000). This is
requires more theoretical and experimental investigation.
performed by replacing u⋆ in eq. (20) with a frequencydependent u′⋆ (f ) defined by
peak are lower with su = 0.4 (TEST441) than in other
2
runs, this is largely due to a reduced feedback of the wave
(u′⋆ )
= u2⋆ (cos θu , sin θu )
Z k Z 2π
age on the wind stress via the τw /τ term in eq. (24).
′
Sin (f , θ)
(cos θ, sin θ) df ′ dθ,
− |su |
C
0
0
3. Consequences of the source term shape
(25)
We have already illustrated the effects of various parameters on spectral shapes in academic time-limited and more
where the sheltering coefficient |su | ∼ 1 can be used to tune
realistic fetch-limited conditions. We now look at real sea
the wind stresses for high winds, which would be largely
states observed in the world ocean. Although wave spectra
overestimated for su = 0. For su > 0 this sheltering is
are difficult to compare to the few available observations,
also applied within the diagnostic tail, which requires the
we have investigated the systematic variation of spectral
estimation of a 3-dimensional look-up table for the high
moments
frequency stress. The shape of the new wind input is ilZ fc
lustrated in figure 7 for fully developped seas. Clearly,
f n E(f )df.
(26)
mn (fc ) =
for relatively young waves the energy levels at the spectral
0
11
with n = 2,3 and 4, and cut-off frequencies in the range
0.2 to 0.4 Hz. Such moments are relevant to a variety of
applications. Ardhuin et al. (2009b) investigated the third
moment, which is proportional to the surface Stokes drift
in deep water, and found that buoy data are very well
represented by a simple function, which typically explains
95% of the variance,
"
1.3 #
0.5
5.9gU10
× 10−4 1.25 − 0.25
m3 (fc ) ≃
(2π)3
fc
min {U10 , 14.5} + 0.027 (Hs − 0.4) ,
6
M odelled
C cu
=−1, su =
0
M odelled
Bidlot et al. (2005)
5
4
mss (%)
×
tions. Although it covers much less data, the analysis of m4
obtained from buoy heave spectra produces results similar
to figure 8.
(27)
H s (m ):
11
10
9
8
7
6
5
4
3
2
1
3
2
where fc is in Hertz, U10 is in meters per second, and Hs
is in meters.
This relationship is well reproduced in hindcasts using
Ccu = −0.4 and su = 1, while the BAJ source terms give a
nearly constant value of m3 when Hs varies and U10 is fixed
(Ardhuin et al. 2009b). Here we also consider the fourth
moment m4 which, for linear waves, is proportional to a
surface mean square slope filtered at the frequency fc . Figure 8 shows that for any given wind speed mssC increases
with the wave height (Gourrion et al. 2002), whereas this
is not the case of m4 in the BAJ parameterization, or,
for very high winds, when Ccu is too strong. In the case
of BAJ, this is due to the (k/kr )2 part in the dissipation
term (eq. 2), which plays a role similar to the cumulative term in our formulation. For Ccu = −1 and su = 0,
the cumulative effect gets too strong for wind speeds over
10 m s−1 , in which case m4 starts to decrease with increasing wave height, whereas for high winds and low (i.e.
young) waves, the high frequency tail is too high and the
mean square slope gets as large as 6%, which is unrealistic.
It thus appears, that the high frequency tail, for su = 0,
responds too much to the wind, hence our use of su = 1 in
the TEST441 combination. The presence of a cumulative
dissipation term allows a different balance in the spectral
regions above the peak, where an equilibrium range with
a spectrum proportional to f −4 develops (Long and Resio
2007), and in the high frequency tail were the spectrum decays like f −5 or possibly a little faster. The spectral level
in the range 0.2 to 0.4 Hz was carefully compared against
buoy data, and was found to be realistic.
These interpretations of the model result assumes that
the high frequency part of the spectrum can be simply converted to a wavenumber spectrum, using linear wave theory. This is not exactly the case as demonstrated by Banner et al. (1989). Also, there is no consensus on the nature
of the spectrum modelled with the energy balance equation
but, since non-resonant nonlinearities are not represented,
the modelled spectra are expected to be more related to
Lagrangian buoy measurements, rather than Eulerian measurements. This matter is left for further studies, together
with a detailed interpretation of altimeter radar cross sec-
0
5
10
15
20
25
5
10
U 1 0 (m /s )
5
O bserved
C−band JASO N 1
15
20
25
U 1 0 (m /s )
0
M odelled
C cu =−0.4, s
u= 1
mss (%)
4
3
2
Fig. 8. Variation of the surface mean square slope estimated as either 0.64/σ0 using the C-band altimeter on
board JASON-1, after the correction of a 1.2 dB bias in
the JASON data, or by integration of modelled spectra
from 0 to 0.72 Hz, with either the Ccu = −0.4 and su = 1
parameterization (TEST441) or the parameterization by
BAJ. For modelled values a constant 0.011 is added to account for the short waves that contribute to the satellite
signal and that are not resolved in the model. This saturated high frequency tail is consistent with the observations
of Vandemark et al. (2004). The original 1 Hz data from
JASON is subsampled at 0.5 Hz and averaged over 10 s,
namely 58 km along the satellite track. The same averaging is applied to the wave model result, giving the 393382
observations reported here, for the first half year of 2007.
4. Verification
In order to provide simplified measures of the difference
between model time series Xmod and observations Xobs we
use the following definitions for the normalized root mean
square error (NRMSE),
sP
2
(Xobs − Xmod )
P 2
(28)
NRMSE(X) =
Xobs
12
the normalized bias,
NB(X) =
sP
Xobs − Xmod
P
,
Xobs
and Pearson’s linear correlation coefficient,
P
Xobs − Xobs Xmod − Xmod
r(X) = qP
2
2 ,
Xobs − Xobs
Xmod − Xmod
(29)
(30)
where the overbar denotes the arithmetic average.
The normalisation of the errors allows a quantitative
comparison between widely different sea state regimes. Because previous studies have often used (non-normalized)
RMSE we also provide RMSE values. In addition to the
coastal fetch-limited case of SHOWEX, presented above,
the parameterizations are calibrated on at the global scale
and validated in two other cases.
a. Global scale results
We present here results for the entire year 2007, using
a stand-alone 0.5◦ resolution grid, covering the globe from
80◦ south to 80◦ north. The model has actually been adjusted to perform well over this data set, but the very large
number of observations (over 2 million altimeter collocation points) makes the model robust, and an independent
validation on 2008 gives identical results. The interested
reader may also look at the monthly reports for the SHOM
model (e.g. Bidlot 2008), generated as part of the model
verification project of the IOC-WMO Joint Commission
on Oceanography and marine Meteorology (JCOMM), in
which the TEST441 parameterization (Ccu = −0.4 and
su = 1) is used, except for the Mediterranean where, the
TEST405 has been preferred for its superior performance
for younger seas. These SHOM models are ran in a combination of two-way nested grids (Tolman 2007). The monthly
JCOMM reports include both analysis and forecasts, but,
since they are produced in a routine setting, many SHOM
calculations from December 2008 to June 2009 have been
affected by wind file transfer problems.
Comparing model results for Hs to well-calibrated (Queffeulou and Croizé-Fillon 2008) altimeter-derived measurements provides a good verification of the model performance in a number of different wave climates. Figure (9)
shows that, as expected, the important positive bias in the
swell-dominated regions when using the BAJ parameterization, has been largely removed. This is essentially the
signature of the specific swell dissipation that is parameterized in Sout . The largest bias pattern now appears in the
southern ocean, reaching 30 cm in the Southern Atlantic.
Although this bias is small compared to the local averaged
wave height, it is rather strange when the model errors are
plotted as a function of wave height in figure (10). Why
13
would the model overestimate the Southern ocean waves
but understimate the very large waves?
The structure of the large bias, also seen in model results with BAJ, is reminiscent of the observed pattern in
iceberg distribution observed by Tournadre et al. (2008).
These observed iceberg distributions are enough to give a
cross-section for incoming waves of the order of 1 to 10% for
a 250 km propagation length. Taking icebergs into account
could actually reverse the sign of the bias. This matter will
be investigated elsewhere.
Also noticeable is a significant negative bias in the equatorial south Pacific, amplified from the same bias obtained
with the BAJ parameterization. It is possible that the
masking of subgrid islands (Tolman 2003) introduces a
bias by neglecting shoreline reflections. This model defect could be exacerbated in this region by the very large
ratio of shoreline length to sea area. This will also require
further investigation. Finally, the negative biases for Hs
on mid-latitude east coasts are reduced but still persist.
It is well known that these areas are also characterized
by strong boundary currents (Gulf Stream, Kuroshio, Agulhas ...) with warm waters that is generally conducive
to wave amplification and faster wind-wave growth (e.g.
Vandemark et al. 2001). Neither effect is included in the
present calculation because the accuracy of both modelled
surface currents and air-sea stability parameterizations are
likely to be insufficient (Collard et al. 2008; Ardhuin et al.
2007).
The reduction of systematic biases clearly contributes
to the reduction of r.m.s. errors, as evident in the equatorial east Pacific (Fig. 11). However, the new parameterization also brings a considerable reduction of scatter,
with reduced errors even where biases are minimal, such as
the trade winds area south of Hawaii, where the NRMSE
for Hs can be as low as 5%. When areas within 400 km
from continents are excluded, because the global model
resolution may be inadequate, significant errors (> 12.5%,
in yellow to red) remain in the northern Indian ocean, on
the North American and Asian east coasts, the Southern
Atlantic. The parameterizations TEST405, TEST437 and
TEST441 produce smaller errors on average than BAJ. It
is likely that the model benefits from the absence of swell
influences on wind seas: swell in BAJ typically leads to a
reduced dissipation and stronger wind wave growth. As
models are adjusted to average sea state conditions, this
adjustement leads to a reduced wind sea growth on east
coasts where there is generally less swell.
Although much more sparse than the altimeter data,
the in situ measurements collected and exchanged as part
as the JCOMM wave model verification is very useful for
constraining other aspects of the sea state. This is illustrated here with mean periods Tm02 for data provided by
the U.K. and French meteorological services, and peak periods Tp for all other sources. It is worth noting that the
Fig. 9. Bias for the year 2007 in centimeters. The global 0.5 WWATCH model is compared to altimeters JASON, ENVISAT and GFO following the method of Rascle et al. (2008). The top panel is the result with the BAJ parameterization,
and the bottom panel is the result with the Ccu = −0.4 and su = 1 (TEST441) parameterization.
14
Fig. 11. Normalized RMSE for the significant wave height over the year 2007, in percents. The global 0.5◦ resolution
WWATCH model is compared to altimeters JASON, ENVISAT and GFO following the method of Rascle et al. (2008).
The top panel is the result with the BAJ parameterization, and the bottom panel is the result with the Ccu = −0.4 and
su = 1 (TEST441) parameterization.
15
dissipation was tested but it tended to deteriorate the results on other parameters. On European coasts, despite a
stronger bias, the errors on Tm02 are particularly reduced.
Again, this reduction of the model scatter can be largely
attributed to the decoupling of swell from windsea growth.
Table 1. Model accuracy for measured wave parameters
over the oceans in 2007. mss data from JASON 1 corresponds to January to July 2007 (393382 co-located points).
Unless otherwise specified by the number in parenthesis,
the cut-off frequency is take to be 0.4 Hz, C stands for
C-band. The normalized bias (NB) is defined as the bias
divided by the r.m.s. observed value, while the scatter index (SI) is defined as the r.m.s. difference between modeled
and observed values, after correction for the bias, normalized by the r.m.s. observed value, and r is Pearson’s correlation coefficient. These global averages are area-weighted,
and the SI and NRMSE are the area-weighted averages of
the local SI and NRMSE.
BAJ TEST TEST TEST
405
437
441
Hs
NB(%)
-2.1
-0.8
0.2
-1.23
SI(%)
11.8
10.5
10.6
10.4
NRMSE(%) 13.0
11.5
11.6
11.3
m4 (C)
NB(%)
-16.1
-4.9
-2.3
-2.5
SI(%)
10.7
9.1
9.1
9.1
r
0.867 0.925
0.931
0.939
Tp
NRMSE(%) 24.1
19.0
19.4
18.2
Tm02
NRMSE(%)
7.6
6.9
6.6
6.7
m3
NB(%)
-14.6
1.7
-2.3
-2.4
SI(%)
20.6
12.6
14.8
12.6
NRMSE(%) 25.3
13.1
13.1
12.8
r
0.934 0.971
0.961
0.973
=−0.4, s
Number of obs.
=−1,
s
6
10
5
10
4
10
3
10
2
10
10
1
Fig. 10. Wave model errors as a function of Hs . All model
parameterizations are used in a global WWATCH model
settings using a 0.5◦ resolution. The model output at 3h
intervals is compared to JASON, ENVISAT and GFO following the method of Rascle et al. (2008). Namely, the
altimeter 1 Hz Ku band estimates of Hs are averaged over
1◦ . After this averaging, the total number of observations
is 2044545. The altimeter estimates are not expected to be
valid for Hs larger than about 12 m, due to the low signal
level and the fact that the waveform used to estimate Hs
is not long enough in these cases.
errors on Hs for in situ platforms are comparable to the
errors against altimeter data.
With the BAJ parameterizations, the largest errors in
the model results are the large biases on peak periods on
the U.S. West coast (Fig. 12), by 1.2 to 1.8 s for most locations, and the understimation of peak periods on the U.S.
East Coast. However, peak and mean periods off the European coasts were generally very well predicted. When using
the TEST441 parameterization, the explicit swell dissipation reduces the bias on periods on the U.S. West coast,
but the problem is not completely solved, with residual biases of 0.2 to 0.4 s. This is consistent with the validation
using satellite SAR data (Fig. 1), that showed a tendency
to underpredict steep swells near the storms and overpredict them in the far field. A simple increase of the swell
The general performance of the parameterizations is
synthetized in table 1. It is interesting to note that the
parameterization TEST405 that uses a diagnostic tail for
2.5 times the mean frequency gives good results in terms
of scatter and bias even for parameters related to short
waves (m3 , m4 ). This use of diagnostic tail is thus a good
pragmatic alternative to the more costly explicit resolution of shorter waves, which requires a smaller adapatative timestep, and more complex parameterizations. The
diagnostic tail generally mimics the effect of both the cumulative and sheltering effects. Yet, the parameterization
TEST441 demonstrates that it is possible to obtain slightly
better results with a free tail. The normalized biases indicated for the mean square slopes are only relative because
16
Fig. 12. Statistics for the year 2007 based on the JCOMM verification data base (Bidlot et al. 2007b). Bais (a,d) and
NRMSE (b,c) for Tp or Tm02 at in situ locations using the BAJ or the proposed TEST441 (Tp is shown at all buoys except
U.K. and French buoys for which Tm02 is shown). The different symbols are only used to help distinguish the various
colors and do not carry extra information.
17
of the approximate calibration of the radar cross section.
They show that the BAJ parameterization (Bidlot et al.
2005), and to a lesser extent the use of a f −5 tail, produce
energy levels that are relatively lower at high frequency.
The KHH parameterization perform well in this simple
wave climate, consistent with prior published applications
with the SWAN model, (Rogers et al. 2003) and RW2007,
without the difficulties discussed above in mixed sea-swell
conditions.
The BAJ, TEST437 and TEST441 models also perform
well here. Taken together with the global comparisons
above, we observe no apparent dependency of model skill
on the scale of the application with these three physics. In
the bias comparison, Fig. 13 top panel, the BAJ model
is nearly identical to the KHH model. Similarly, the two
new models are also very close. Although TEST441 and
TEST437 produce a minor underestimation of Hs (Table
2), they give slightly better correlations with observed wave
heights and mean periods.
b. Lake Michigan
At the global scale, the sea state is never very young,
and it is desirable to also verify the robustness of the parameterization in conditions that are more representative
of the coastal ocean. We thus follow the analysis of wave
model performances by Rogers and Wang (2007), hereinafter RW2007, and give results for the Lake Michigan,
representative of relatively young waves. The model was
applied with parameterizations BAJ, TC, TEST437 and
TEST441 over the same time frame as investigated by
RW2007, September 1 to November 14, 2002. The model
setting and forcing fields are identical to the one defined
by (Rogers et al. 2003), with a 2 km resolution grid, a 10◦
directional resolution, and a wind field defined from in situ
observations. The results at the position of National Data
Buoys Center’s buoy 45007 are compared to corresponding
measurements.
Using the directional validation method proposed by
these authors, the TC parameterization understimates the
directional spread σθ by 1.2 to 1.6◦ in the range 0.8 to 2.0
fp , and more at higher frequencies. The understimation
with BAJ is about half, and the TEST441 and TEST437
overpredict the directional spread by about 2.3 to 5.9◦ in
the range 0.8 to 2.0fp , and less so for higher frequencies.
It thus appears that the broadening introduced to fit the
SHOWEX 1999 observations is not optimal for other situations. A similar positive bias on directional spread is also
found in global hindcasts.
Further results are presented in Table 2 and Figure 13.
The top panel of Figure 13 compare the summed values
of co-located model and observed spectral densities for the
duration of the simulation. This presentation provides frequency distribution of bias of the various models, while also
indicating the relative contribution of each frequency to the
wave climate for this region and time period. The lower
panel shows the correlation coefficient r for the equivalent
significant waveheights computed for multiple frequency
bands. This is presented in terms of f /fp (bin width=0.1),
with fp being calculated as the stabilized “synthetic peak
frequency” of the corresponding buoy spectrum, as defined
in RW2007.
The most noticeable outcome of these comparisons is
the relatively poor performance of the TC parameterizations. Taken in context with other TC results presented
herein and prior undocumented application of the model in
the Great Lakes with model wind fields, this suggests that
these parameterizations have some undesirable dependence
on scale, with the parameters adjusted by Tolman (2002b)
being most optimal for ocean-scale applications.
Table 2. Model-data comparison at NDBC buoy 45007
(Lake Michigan) for four frequency bands. Statistics are
given for equivalent significant wave heights in the bands
0.5fp < f < 0.8fp (band 1), 0.8fp < f < 1.2fp (band 2),
1.2fp < f < 2fp (band 3) 2fp < f < 3fp (band 4). The
KHH n = 2 run corresponds to the dissipation parameterization defined by (Rogers et al. 2003) based on (Komen
et al. 1984) and applied in the WW3 code. For the peak
frequency band, statistics are also given for the directional
spread σθ .
BAJ
TC
TEST TEST KHH
437
441
n=2
Hs band 1
SI(%)
76
71
77
82
76
Hs band 2
SI(%)
16
19
15
15
15
Hs band 3
SI(%)
18
26
17
17
19
Hs band 4
SI(%)
20
32
18
17
22
σθ band 2
SI(%)
22
24
30
30
25
bias (◦ )
-0.4
-1.6
2.3
2.6
0.8
Hs bias (m) -0.03 -0.11 -0.02
-0.02
0.04
r
0.95
0.94
0.96
0.96
0.96
SI(%)
19
25
18
18
19
Tm02
bias (s)
0.02
-.39
-0.05
-0.05
0.01
r
0.89
0.87
0.90
0.90
0.90
SI(%)
10
14
9
9
10
It thus appears that for such young seas, the directional
spreading of the parameterization could be improved, but
the energy content of various frequency bands, and as a
result the mean period, are reproduced with less scatter
than with previous parameterizations.
18
Σ F(f) (m2/Hz)
10
10
10
in the wave model integration, and in any event, only affect
weaker wind seas far from the storm center.
Bathymetry is taken from the Naval Research Laboratory’s 2 minute resolution database, DBDB2, coarsened to
the model grid resolution (0.1 deg). The directional resolution is 10◦ , and the frequency range is limited to 0.04180.4117 Hz. The model was applied from September 13 to
September 16 2004. Model results are illustrated by figure
14. The models were validated at all the buoys in the gulf
of Mexico. Results from buoy 42040, where waves where
largest, are shown here.
3
2
buoy
BAJ
TC
TEST 437
TEST 441
KHH n=2
1
0.1
0.15
0.2
0.25
freq (Hz)
0.3
0.35
0.4
1.4
1.6
1.8
2
r
1
0.5
0
0.8
1
1.2
f/fp
Fig. 13. Model-data comparison at NDBC buoy 45007
(Lake Michigan). Top panel: Comparison of summed spectral density versus frequency for the duration of the simulation. Lower panel: Correlation coefficients versus normalized frequency (see text for explanation).
c. Hurricane Ivan
Although the global hindcast does contain quite a few
extreme events, with significant wave heights up to 17 m,
these were obtained with a relatively coarse wave model
grid and wind forcing (0.5◦ resolution and 6-h timestep)
that is insufficient to resolve small storms such as tropical
cyclones (Tolman and Alves 2005). Hurricane waves do
share many similarities with more usual sea states (Young
2005), but the high winds and their rapid rotation are particularly challenging for numerical wave models. It is thus
necessary to verify that the new source functions perform
adequately under extreme wind conditions. A simulation
of Hurricane Ivan (Gulf of Mexico, September 2004) is chosen for this purpose because it was extensively measured
(e.g. Wang et al. 2005) and hindcasted.
Winds for this simulation are based on gridded surface
wind analyses created by NOAA’s Hurricane Research Division (HRD). These analyses are at three hour intervals,
which for a small, fast-moving weather system is temporally too coarse to provide directly to the wave model.
Therefore, as an intermediate step, fields are reprocessed
to 30 minute intervals, with the storm position updated at
each interval (thus, semi-Lagrangian interpolation). The
wind speeds are reduced by factor 1/1.11 to convert from
maximum sustained gust to hourly mean. The HRD winds
do not cover the entire computational domain. For areas
falling outside the domain, the nearest NDBC buoy wind
observation is used. This produces some non-physical spatial discontinuities in the wind field, but these are smoothed
Fig. 14. Time series of model and buoy significant wave
height (top) and partial wave height (bottom, for f < 0.06
Hz), during the passage of Hurrican Ivan, at buoy 42040.
The 10-m discus buoy was capized by waves and could not
record waves after 16 September.
Model runs with parameterizations BAJ, TEST437 and
TEST441 give very similar results: close to the observations, except for the highest waves (Hs > 13 m at buoy
42040) where TEST437 and TEST441 give slightly smaller
values. Results with the TC parameterization are generally
lower in terms of Hs than all these parameterizations that
share a Janssen-type input. It appears that the new source
term perform similarly to BAJ and are able to reproduce
such young waves and severe sea states.
Because the wind forcing enters the wave model through
a wind stress parameterized in a way that may not apply
to such conditions, it is worthwhile to re-examine some
choices made above. In particular the surface roughness
was allowed to exceed 0.002 in TEST441b, which resulted
in better estimates of Hs . Lifting this constraint shows
that, for these very high winds, the wind sheltering effect plays a similar role to the limitation of the roughness
to z0,max , with the difference that it tends to narrow the
wind input spectrum (Fig. 7). This narrower wind input
19
has a limited effect of the wind stress and Hs , but is has a
noticeable effect on the spectral shape. This is illustrated
by the low frequency energy that appears to be strongly
overestimated before the peak of the storm for TEST441b.
In general the new parameterizations provide results that
are as reasonable as those of previous parameterizations,
given the uncertainty of the wind forcing.
5. Conclusions
A set of parameterization for the dissipation source
terms of the wave energy balance equation have been proposed, based on known properties of swell dissipation and
wave breaking statistics. This dissipation includes an explicit nonlinear swell dissipation and a wave breaking parameterization that contains a cumulative term, representing the dissipation of short waves by longer breakers, and
different dissipation rates for different directions. These
dissipation parameterizations have been combined with a
modified form of the wind input proposed by Janssen (1991),
in which the questionable gustiness parameter zα has been
reduced, and the general shape of the wind input has been
significantly modified. The resulting source term balance is
thus markedly different from the previous proposed forms,
with a near-balance for very old seas between the air-sea
friction term, that dissipates swell, and the nonlinear energy flux to low frequencies. Also, the wind input is concentrated in a narrower range of frequencies.
For younger seas the wind input is relatively weaker
than given by Janssen (1991) but stronger than given by
Tolman and Chalikov (1996) (TC). However, the dissipation at the peak is generally stronger because it is essentially based on a local steepness and these dominant waves
are the steepest in the sea state. As a result the short
fetch growth is relatively weaker than with the source term
combination proposed by Bidlot et al. (2007a) (BAJ). The
choice of parameters tested here tend to produce broader
directional spectra than observed in the Lake Michigan and
global hindcasts, and slanting fetch directions that are too
oblique relative to the wind (Fig. 2). In this respect the
new source terms are intermediate between BAJ and TC.
Another likely defect comes from the definition of the
saturation level used to define the breaking-induced dissipation. Here, as in the work by van der Westhuysen et al.
(2007), the saturation is local in frequency space, whereas
wave breaking is naturally expected to have a relatively
broad impact due to its localization in space and time
(Hasselmann 1974). This is expected to produce an overestimation of the energy just below the peak, and an understimation at the peak of the saturation spectrum. These
effects likely contribute to the persistent overestimation of
low frequency energy in the model.
In spite of these defects, the new parameterization produces robust results and clearly outperform the Bidlot et al.
20
(2007a) parameterization in global hindcasts, whether one
considers dominant wave parameters, Hs , Tm02 and Tp or
parameters sensitive to the high frequency content, such as
the surface Stokes Uss drift or the mean square slope. At
global scales, errors on Hs , Tp and Uss are - on average - reduced by 15, 25 and 50% relative to those obtained with the
parameterization by Bidlot et al. (2007a). All global hindcasts results, for the year 2007 at least, are available for further analysis at ftp://ftp.ifremer.fr/ifremer/cersat/products/gridded/
.
Another important aspect was the validation at regional
to global scales. We note that the TC paper does include
verification with steady-state, fetch-limited growth curves.
Though such verification is a useful step, the outcome of
the Lake Michigan hindcast suggests that such verification
does in no way anticipate skill in real sub-regional-scale
applications. One of the parameterizations proposed here
(TEST441) also gives slightly lower performance for young
seas, which is not obvious in the case of lake Michigan,
but was revealed by hindcasts of Mediterranean waves (not
shown).
Because our intention was only to demonstrate the capability of new dissipation parameterizations and the resulting source term balances, we have not fully adjusted
the 18 parameters that define the deep water parameterizations, compared to about 9 with Bidlot et al. (2007a).
The results presented here are thus preliminary in terms of
model performance, which is why the parameterizations are
still given temporary names like TEST441. As illustrated
by the hurrican Ivan hindcast, some parameters, such as
z0,max , is probably unnecessary: in that particular case the
removal of z0,max improved the results, but for global scale
results it had no impact at all (not shown).
Because 5 of the extra parameters define the air-sea
friction term that produces swell dissipation, and 2 define the cumulative breaking term, it is feasible to define
a systematic adjustement procedure that should produce
further improvements by separately adjusting swell, wind
sea peak, and high frequency properties. In particular the
directional distribution may be improved by making the
dissipation term more isotropic (i.e. taking δd > 0.3) or
modifying the definition of the saturation parameter B ′ in
equation (12). In part II we shall further investigate the
response of the wave field to varying currents, from global
scales to regional tidal currents. It is particularly expected
that wave steepening will produce much more dissipation
due to breaking, as envisaged by Phillips (1984).
Obviously, it is well known that the Discrete Interaction Approximation used here to compute the non-linear
interactions is the source of large errors, and further calculations, will be performed using a more accurate estimation
of these interactions in part III.
Ardhuin, F., F. Collard, B. Chapron, P. Queffeulou, J.-F.
Filipot, and M. Hamon, 2008a: Spectral wave dissipation
based on observations: a global validation. Proceedings
of Chinese-German Joint Symposium on Hydraulics and
Ocean Engineering, Darmstadt, Germany, 391–400.
Acknowledgments.
This research would not have been possible without the
dedication of Hendrik Tolman, Henrique Alves, and Arun
Chawla in putting together the core of the WAVEWATCHIIITM code. Wind and wave data were kindly provided by
ECMWF, Météo-France, and the French Centre d’Etudes
Techniques Maritimes Et Fluviales (CETMEF). The SHOM
buoy deployments were managed by David Corman with
precious help from Guy Amis. Wave data were kindly provided by the U.S. Navy, ESA and CNES, and the many in
situ contributors to the JCOMM (WMO-IOC) exchange
program coordinated by Jean-Raymond Bidlot.
Ardhuin, F., T. H. C. Herbers, K. P. Watts, G. P. van Vledder, R. Jensen, and H. Graber, 2007: Swell and slanting
fetch effects on wind wave growth. J. Phys. Oceanogr.,
37, doi:10.1175/JPO3039.1, 908–931.
Ardhuin, F. and A. D. Jenkins, 2005: On the effect of
wind and turbulence on ocean swell. Proceedings of the
15th International Polar and Offshore Engineering Conference, June 19–24, Seoul, South Korea, ISOPE, volume III, 429–434.
APPENDIX A
Parameter settings
— 2006: On the interaction of surface waves and upper
ocean turbulence. J. Phys. Oceanogr., 36, 551–557.
All parameters defining the dissipation source function
and their numerical values are listed in table 3 for the windwave interaction term Satm and table 4 for the wave-ocean
interaction term Soc . We also recall that the nonlinear coupling coefficient (variable NLPROP in WWATCH) is set to
2.78×107 in all cases, except for the two parameterizations
mostly used here, with Cnl = 2.5 × 107 in TEST437 and
TEST441. Although the best performance for most parameters is obtained with the TEST441 settings, its underestimation of extreme sea states may be a problem in some
applications for which the TEST437 may be preferred. A
full tuning of the model has not been tried yet and it is
possible that a simple adjustment of βmax Ccu , rcu and su
may produce even better results. Finally, these parameters have been mostly adjusted for deep water conditions
using ECMWF winds. Using other sources of winds for
large scale applications may require a retuning of the wind
source function, which can be best performed by a readjustment of βmax .
Ardhuin, F. and A. Le Boyer, 2006: Numerical modelling
of sea states: validation of spectral shapes (in French).
Navigation, 54, 55–71.
Ardhuin, F., L. Marié, N. Rascle, P. Forget, and A. Roland,
2009b: Observation and estimation of Lagrangian,
Stokes and Eulerian currents induced by wind and waves
at the sea surface. J. Phys. Oceanogr., 39, 2820–2838.
Ardhuin, F., N. Rascle, and K. A. Belibassakis, 2008b:
Explicit wave-averaged primitive equations using a
generalized Lagrangian mean. Ocean Modelling, 20,
doi:10.1016/j.ocemod.2007.07.001, 35–60.
Babanin, A., K. Tsagareli, I. Young, and D. Walker, 2007:
Implementation of new experimental input/dissipation
terms for modelling spectral evolution of wind waves.
Proceedings of the 10th International workshop on wave
hindcasting and forecasting, Oahu, Hawaii.
Babanin, A., I. Young, and M. Banner, 2001: Breaking
probabilities for dominant surface waves on water of finite depth. J. Geophys. Res., 106, 11659–11676.
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25
Parameter
α0
βmax
zα
z0,max
su
s0
s1
s2
s3
0.5Rec Hs
Cdsv
rz0
zu
p
see eq.
(20)
(21)
(21)
(23)
(25)
(10)
(10)
(10)
(8)
(11)
(22)
(20)
variable in code
ALPHA0
BETAMAX
ZALP
Z0MAX
TAUWSHELTER
SWELLFPAR
SWELLF
SWELLF2
SWELLF3
SWELLF4
SWELLF5
Z0RAT
ZWND
SINTHP
WAM-Cycle4
0.01
1.2
0.011
N.A.
0.0
0
0.0
0.0
0.0
0.0
0.0
0.0
10
2
BAJ
0.0095
1.2
0.011
N.A.
0.0
0
0.0
0.0
0.0
1 × 105
0.0
0.0
10
2
TEST405
idem
1.55
0.006
0.002
0.0
3
0.8
-0.018
0.015
idem
1.2
0.04
10
2
TEST437
idem
1.52
idem
idem
0.0
idem
idem
idem
idem
idem
idem
idem
10
2
TEST441
idem
idem
idem
idem
1.0
idem
idem
idem
idem
idem
idem
idem
10
2
Table 3. Wind-wave interaction parameters as implemented in version 3.14-SHOM of the WAVEWATCH III code, and
values used in the tests presented here. In WWATCH, all parameters are accessible via the SIN3 namelist. All of these
parameters are included in version 3.14 of WWATCH. s0 is a switch that, if nonzero, activates the calculation of Sout .
Parameter
Cds
r
fFM
δ1
δ2
sat
Cds
Clf
Chf
∆θ
Br
rcu
2 ∗ Ccu
sB
δd
M0
M1
M2
M3
M4
see eq.
(2)
(2)
(7)
(2)
(2)
(13)
(12)
(12)
(18)
(18)
(12)
(13)
variable in code
SDSC1
WNMEANP
FXFM3
SDSDELT
SDSDELTA2
SDSC2
SDSLF
SDSHF
SDSDTH
SDSBR
SDSBRF1
SDSC3
SDSCOS
SDSDC6
SDSBM0
SDSBM1
SDSBM2
SDSBM3
SDSBM4
WAM4
-4.5
-0.5
2.5
0.5
0.5
0.0
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
BAJ
-2.1
0.5
2.5
0.4
0.6
0.0
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
TEST405
0.0
0.5
2.5
0.0
0.0
−2.2 × 10−5
0.0
0.0
80
1.2 × 10−3
0.0
0.0
0.0
1.0
1.0
0.2428
1.9995
-2.5709
1.3286
TEST437
idem
idem
9.9
idem
idem
idem
idem
idem
idem
9 × 10−4
0.5
-2.0
2.0
0.3
idem
idem
idem
idem
idem
TEST441
idem
idem
idem
idem
idem
idem
idem
idem
idem
idem
idem
-0.8
idem
0.3
idem
idem
idem
idem
idem
TEST443
idem
idem
idem
idem
idem
idem
idem
idem
idem
idem
idem
idem
idem
0.0
idem
idem
idem
idem
idem
Table 4. Dissipation parameter as implemented in version 3.14-SHOM of the WAVEWATCH III code, and values used
in the tests presented here. In WWATCH, all parameters are accessible via the SDS3 namelist. The only parameters not
defined in the present paper are Clf and Chf which act like switches to activate the BAJ parameterization for the part
of the spectrum with saturation below and abovce the spectrum, respectively. Most of these are also included in version
3.14, except for Ccu which is needed for the TEST437 and TEST441 with results described here. The TEST405 can be
ran with version 3.14. The parameter M0 is a switch for the correction or not of Br into Br′ , when M0 = 1, as is the case
here, the correction is not applied.
26